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Abstract

We introduce a novel approach of stabilizing the dynamics of excitation waves by spatially
extended sub-threshold periodic forcing. Entrainment of unstable primary waves has
been studied numerically for different amplitudes and frequencies of additional sub-threshold
stimuli. We determined entrainment regimes under which excitation blocks were transformed
into consistent 1:1 responses. These responses were spatially homogeneous and synchronized
in the entire excitable medium. Compared to primary pulses, pulses entrained by secondary
stimulations were stable at considerably shorter periods which decreased at higher
amplitudes and greater number of secondary stimuli. Our results suggest a practical
methodology for stabilization of excitation in reaction-diffusion media such as nerve
tissue with regions of reduced excitability.

Background

Dynamics of excitation waves in reaction-diffusion media can be altered by spatio-temporal
periodic forcing. Additional (secondary) periodic stimulations superimposed on primary
forcing may alter the primary excitation waves and entrain (lock) them to the period
of secondary stimuli. Locking of primary waves to the period of secondary stimulations
occurs at particular values of forcing periods and amplitudes. This resonant shift
is characterized by Arnold tongues which determine the margins of different types
of M : N (M ≥ N, N ≥1) locking responses as a function of amplitude and frequency of external forcing
[1,2].

It was found that the phenomenon of locking manifests itself in different ways depending
on the spatio-temporal complexity of the particular reaction-diffusion system. For
example, under the periodic forcing, spatially uniform two-dimensional BZ reaction
oscillations were transformed into standing wave type labyrinths of complex geometry
[1]. It was also demonstrated that one-dimensional Turing patterns can be modulated by
spatio-temporal forcing in the form of a travelling wave [3].

A similar type of resonant behaviour of Turing patterns was observed in experiments
with photosensitive chemical reactions [4]. It was also shown that the presence of large amplitude periodic forcing in a one-dimensional
bistable reaction-diffusion medium resulted in bifurcating of an originally stable
wavefront into two counter propagating wavefronts [5]. In contrast, modulating the intensity and frequency of the periodic forcing controlled
the trajectory and rotational frequency of two-dimensional spiral waves in the Oregonator
model [6].

Additional periodic forcing can be also applied for locking of primary waves in some
biological excitable media. A typical example of practical realization of such resonant
dynamics is a post-traumatic adjustment of excitation in nerves with impaired excitability.
It was shown that impaired excitation may be restored by applying additional functional
electrical stimulation using implantable [7] or body surface stimulation electrodes [8-10]. This method has been confirmed as an effective tool for restoration of movement
of paralyzed muscles in individuals with a variety of neurological impairments [11].

Usually, after severe neuromuscular injuries nerve conductivity is significantly reduced,
which in turn, prevents the passage of excitation waves through neuromuscular transmitters.
Under these circumstances propagation of excitation pulses is marginally stable and
implementation of functional electrical stimulation necessitates a significant increase
of frequencies and amplitudes of additional electrical stimuli. The latest, instead
of stabilization of propagation, can facilitate conduction blocks and may completely
disrupt the process of training paralyzed muscles.

To address this deficiency we investigate a new approach of entrainment of marginally
stable excitation waves by spatially extended low amplitude sub-threshold forcing.
We demonstrate that such sub-threshold forcing can transform excitation blocks to
stable 1:1 responses synchronized in the entire excitable medium.

Methods

Dynamics of conduction in nerves is simulated by Fitzhugh-Nagumo type reaction-diffusion
equations with just two fundamental excitation and post-excitation recovery variables
[12,13]. Unlike our recent work where we studied the dynamics initiated by a single excitation
source [14], the adjusted model is set up to reflect the interference of several sources which
deliver multiple stimuli of different amplitudes. In particular, the main source delivered
localized over-threshold stimuli applied at the beginning of the cable and a set of
additional sources originated secondary sub-threshold forcing which was extended throughout
the entire cable (Eqs. 1).

(1)

Here u and v are dimensionless excitation and recovery functions, respectively. ε is a small parameter
and vr is the excitation threshold. λ and ζ control the rates of changes of excitation and
recovery functions, respectively.

The system of Eqs. 1 was solved numerically in a one-dimensional cable of finite length
using a second order explicit difference scheme with zero flux boundary conditions
[15]. Spatial, Δx, and temporal, Δt, steps used in the numerical integration were equal to 0.23 and 0.0072, respectively.
The cable length, L, was equal to 150Δx. Primary forcing, P (x, t), was a train of rectangular pulses with duration 100Δt, an over-threshold amplitude A0 and period T0. Primary stimuli were applied near the left end of the cable between x = 2Δx and x = 15Δx. Secondary forcing {S (x - xi, t), i = 1,2,..., n} also a rectangular pulse train with duration 100 Δt, was delivered at n equidistant locations between x = 40Δx and x = 140Δx. The secondary stimuli were simultaneously activated after the wavefront which resulted
from the first primary stimulation arrived at the end of the cable. The amplitude
and period of secondary stimulations were equal to A and T, respectively.

Unless mentioned otherwise, values of parameters in all computations were ε = 0.1, λ = 0.4, ζ = 1.2, A0 = 1.4, α = 0.31, β = 0.0025 and n = 6. As in [14] we used a simplified primary rate dependent excitation threshold given by linear
equation vr = α - βT0, α > 0, β > 0. The duration of a pulse, Th, was measured as the time interval between consecutive intersections of u and v near their rest and excited states, respectively (Figure 1). The steady state value of Th was computed after 80 primary stimulation periods.

Figure 1.Excitation, u, (solid line) and recovery, v, (dashed line) variables at x = L /2 for the short steady state pulse. Left and right intersections between u and v mark the beginning and the end of the pulse, respectively.

Results

In the absence of secondary stimulation (A = 0), at long periods T0 the variable v has enough time to reach its steady state vr before the next stimulus is applied. However, as T0 is reduced to a critical limit, Tend, the system does not respond to every stimulus and exhibits unstable M : N (M > N ) excitation blocks which occur due to incomplete recovery of control variable v. Figure 2 compares phase portraits of the system at two values of T0. At T0 = 60, u closely follows its nullcline and v almost completely recovers to its steady state value of 0.16 as depicted by the intersection
of the u - v nullclines. However, at T0 = Tend = 30, deviations of u and v from their nullclines are quite significant, thereby v recovers to a value which is much higher than the corresponding threshold of 0.23.

Figure 2.Phase portraits of the system at x = L/2 for T0 = Tend = 30 (dashed contour) and T0 = 60 (solid contour). Nullcline of u is given by thick N-shaped line. Dashed and solid lines with intercepts at vr = 0.23 and vr = 0.16 are nullclines of v for T0 = 30 and T0 = 60, respectively.

Analysis of pulse duration Th at the critical primary period Tend = 30, as a function of secondary frequency F = T -1 and forcing amplitude A ≥ 0 revealed a variety of entrainment regimes (Figure 3). We found that the system did not respond to secondary stimuli at amplitudes which
were smaller than critical values depicted by the curve with circular markers. For
the amplitudes above this curve and frequencies smaller than F0 we observed intermediate M : M responses with M greater than one.

For even greater amplitudes, above the upper curve with square markers, the system
locked to secondary stimuli with consistent 1:1 responses. It should be noticed that
such locking occurred over a wide range of secondary frequencies, and that secondary
amplitudes were five times smaller than the amplitudes of primary stimulations. We
also observed that for frequencies greater than F0 the entrainment of blocked excitation occurred without intermediate M:M responses (Figure 3).

Spatio-temporal contours of u shown in Figure 4 demonstrate expected unstable responses to primary stimulation at T0 < Tend. Temporal dynamics of u and v, as well as spatio-temporal contours of u, show 3:2 excitation blocks (Figure 4,5). However, in the presence of secondary stimulations such unstable responses can
be entrained and stabilized by secondary driving even at T0 < Tend. Indeed, Figure 6 demonstrates that 3:2 blocks can be transformed into stable 1:1 responses which evolve
homogeneously in the entire cable except for short segments located near the site
of primary stimulation.

Figure 4.Gray-scale patterns in the spatio-temporal evolution of u. Corresponding scale bar is shown on the right.

Formation of these fully synchronized responses are preceded by very short (~ 0.02T0) transient periods during which standing wave type oscillations of u rapidly saturate at constant excitation levels (Figure 7).

Figure 7.Temporal evolution of u near the bottom of the second band as depicted in Fig. 6. Profiles are shown for three equidistant moments of time, starting at tinit = 136, δt = 0.36.

Stabilization of the system due to secondary driving can also be achieved using a
greater number of secondary stimulation sources. Correspondingly, Figure 10 demonstrates that an initial two-fold increase of the number of secondary sources
from 2 to 4 extends the region of stability towards shorter values of Tend. Further increase of the number of sources saturates these changes at progressively
shorter values of Tend for smaller coefficients β.

Conclusions

In summary, we demonstrated that additional sub-threshold driving stimuli can entrain
otherwise unstable primary reaction-diffusion waves and transform M : N excitation blocks into stable 1:1 spatially homogeneous responses synchronized in
the entire cable. Compared to pulses resulting from primary forcing alone, pulses
entrained by secondary stimulations were stable at considerably shorter periods. These
periods decreased at higher amplitudes and greater number of secondary stimuli. We
also found that locking to stable 1:1 responses occurred over a wide range of secondary
frequencies. In addition, the sub-threshold secondary amplitudes were a factor of
five smaller than the amplitudes of primary stimuli. Our results outline the possibility
of entrainment of reaction-diffusion waves by sub-threshold additional driving and
may be applied for stabilization of excitation in nerves with regions of impaired
excitability [16].

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

JS conceived, developed and designed the theory and numerical experiments. VV performed
the numerical simulations and has been involved in drafting and revising the manuscript.
Both authors read and approved the final manuscript.

Acknowledgements

We would like to thank Vladimir Polotski and Shyam Aravamudhan for useful discussions
and continuous interest to our work. We also thank Alan Covell for editorial comments.